A108211 a(n) = 16*n^2 + 1.

17, 65, 145, 257, 401, 577, 785, 1025, 1297, 1601, 1937, 2305, 2705, 3137, 3601, 4097, 4625, 5185, 5777, 6401, 7057, 7745, 8465, 9217, 10001, 10817, 11665, 12545, 13457, 14401, 15377, 16385, 17425, 18497, 19601, 20737, 21905, 23105, 24337, 25601, 26897, 28225, 29585

Offset: 1

Author

Reinhard Zumkeller, Jun 15 2005

Keywords

• nonn
• easy

Comments

Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.

Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - Jonathan Vos Post, Jun 18 2005

The identity (16*n^2 + 1)^2 - (64*n^2 + 8)*(2*n)^2 = 1 can be written as a(n)^2 - A158488(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012

Sequence found by reading the line from 17, in the direction 17, 65, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Conjecture: a(n) = floor(1/(1/(4*n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2*n))). - Clark Kimberling, Sep 09 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Maltese Cross.
• Eric Weisstein's World of Mathematics, Gaullist Cross.
• Eric Weisstein's World of Mathematics, Greek Cross.
• Eric Weisstein's World of Mathematics, Latin Cross.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A005843, A016802, A053755, A074377, A158488.

Programs

Magma

I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012

Maple

A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # Wesley Ivan Hurt, Sep 24 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
16*Range[50]^2+1 (* Harvey P. Dale, Jun 06 2025 *)

PARI

a(n)= 16*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011

Formula

a(n) = A002522(4*n) = A016802(n) + 1.

G.f.: x*(17+14*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 08 2012

From Amiram Eldar, Jul 13 2020: (Start)

Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).

Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)

From Elmo R. Oliveira, Jan 17 2025: (Start)

E.g.f.: exp(x)*(16*x^2 + 16*x + 1) - 1.

a(n) = A053755(2*n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)